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Graphs of Sec, Cosec and Cot Functions | sec、cosec、cot 函数图像

2026-08-30 18:56 · 本文部分内容由AI辅助创作

📚 Graphs of Sec, Cosec and Cot Functions | sec、cosec、cot 函数图像

Understanding the graphs of the reciprocal trigonometric functions is essential for solving equations, sketching transformations, and analysing periodic behaviour in A-level Mathematics.

理解余割、正割和余切这三个倒数三角函数的图像，是A-level数学中解方程、画变换图像和分析周期行为的重要基础。

1. Definitions and Basic Relationships | 定义与基本关系

The secant, cosecant and cotangent functions are defined as reciprocals of the three primary trigonometric functions.

正割函数、余割函数和余切函数分别是三个基本三角函数的倒数。

an identities that connect these functions: 1 + tan² x = sec² x and 1 + cot² x = cosec² x.

还需要记住联系这些函数的勾股恒等式：1 + tan² x = sec² x 和 1 + cot² x = cosec² x。

2. The Graph of y = sec x | y = sec x 的图像

is best understood by starting from y = cos x and taking reciprocals of the y-coordinates.

画 y = sec x 的图像最好从 y = cos x 出发，对每个 y 坐标取倒数。

Key features of y = sec x:

y = sec x 的关键特征：

proaches π/2 from the left, cos x approaches 0 from positive values, so sec x tends to +∞.

例如，当 x = 0 时，cos 0 = 1，所以 sec 0 = 1。当 x = π/3 时，cos(π/3) = 1/2，所以 sec(π/3) = 2。当 x 从左侧接近 π/2 时，cos x 从正值接近 0，所以 sec x 趋向 +∞。

3. The Graph of y = cosec x | y = cosec x 的图像

The graph of y = cosec x is obtained from y = sin x in the same reciprocal manner.

y = cosec x 的图像由 y = sin x 以相同的倒数方式得到。

Key features of y = cosec x:

y = cosec x 的关键特征：

as asymptotes at these points and consists of separate U-shaped branches alternating above and below the x-axis.

注意因为 sin x 在 x = 0、±π、±2π 等处为零，余割图像在这些点有渐近线，并由在 x 轴上下交替的分离的 U 形分支组成。

4. The Graph of y = cot x | y = cot x 的图像

The cotangent function is different in nature: it is the reciprocal of tan x, but it can also be expressed as cos x / sin x.

余切函数性质不同：它是 tan x 的倒数，但也可以表示为 cos x / sin x。

Key features of y = cot x:

y = cot x 的关键特征：

Unlike sec and cosec, cot x has no turning points and its graph is continuous between asymptotes.

与 sec 和 cosec 不同，cot x 没有极值点，并且其图像在渐近线之间是连续的。

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5. Comparing Periods and Ranges | 周期与值域对比

The table below summarises the essential features of the three functions.

下表总结了这三个函数的基本特征。

Function Period Asymptotes Range
sec x 2π x = π/2 + nπ (-∞, -1] ∪ [1, ∞)
cosec x 2π x = nπ (-∞, -1] ∪ [1, ∞)
cot x π x = nπ (-∞, ∞)

Note that sec x and cosec x share the same period and range, but their asymptotes and turning points are shifted by π/2. Meanwhile cot x has a shorter period and a full real-number range.

注意 sec x 和 cosec x 有相同的周期和值域，但它们的渐近线和极值点相差 π/2。而 cot x 的周期更短且值域为全体实数。

6. Sketching Transformed Graphs | 画变换后的图像

Transformations of sec, cosec and cot functions follow the same rules as other functions: f(x + a), f(x) + a, a f(x), and f(ax) cause translations, vertical stretches, and horizontal stretches/compressions.

sec、cosec 和 cot 函数的变换遵循与一般函数相同的规则：f(x + a)、f(x) + a、a f(x) 和 f(ax) 分别产生平移、纵向伸缩和横向伸缩。

For example, to sketch y = 2 sec x - 1:

例如，要画 y = 2 sec x - 1：

7. Solving Equations Using Graphs | 用图像解方程

f reciprocal trig functions are often used to solve equations such as sec x = 2 or cot x = √3 within a given interval.

倒数三角函数的图像常用于解形如 sec x = 2 或 cot x = √3 在给定区间内的方程。

Example: Solve sec x = 2 for 0 ≤ x ≤ 2π.

例：解方程 sec x = 2，其中 0 ≤ x ≤ 2π。

simpler equation. In the interval [0, 2π], cos x = 1/2 has solutions x = π/3 and x = 5π/3.

因为 sec x = 2 意味着 cos x = 1/2，我们解这个更简单的方程。在区间 [0, 2π] 内，cos x = 1/2 的解为 x = π/3 和 x = 5π/3。

unction corresponds to the same x-values as solving the corresponding primary equation, with careful attention to domain restrictions.

一般来说，用水平线 y = k 与倒数函数图像相交所得的 x 值，与解相应基本方程所得一致，但需注意定义域限制。

8. Asymptotes and the Limiting Behaviour | 渐近线与极限行为

ec x, asymptotes occur where cos x = 0; for cosec x and cot x, they occur where sin x = 0.

理解渐近线对正确画图至关重要。对于 sec x，渐近线出现在 cos x = 0 处；对于 cosec x 和 cot x，它们出现在 sin x = 0 处。

ote, the function value tends to either +∞ or -∞ depending on the sign of the denominator:

在渐近线附近，函数值趋向 +∞ 或 -∞，具体取决于分母的符号：